#232157

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19745	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

503 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,10,12,7)(4,13,11,16)(5,8)(6,9)(14,15)(17,18)(19,24,29,32)(20,30,28,31)(21,23)(22,26)(25,27);(1,3,6,12)(2,7,9,10)(4,14,11,5)(8,16,15,13)(18,23)(19,20)(21,27)(25,26)(28,29)(30,31);(1,4,5,3,6,11,14,12)(2,7,15,16,9,10,8,13)(17,19,27,20,22,29,21,28)(18,24,25,30,26,32,23,31);(2,8,9,15)(7,13,10,16)(17,20,21,19,22,28,27,29)(18,24,25,30,26,32,23,31)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ef,d^2=e,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245048, #72245222, #72245267
maximal subgroup of	#72245048, #72245124, #72245221, …
maximal normal subgroup of	#72245048, #72245124, #72245221, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139353);